“Russia, if you’re listening, I hope you’re able to find the 30,000 emails that are missing,” Mr. Trump said, staring directly into the cameras. “I think you will probably be rewarded mightily by our press.”

Mr. Trump’s call was an extraordinary moment at a time when Russia is being accused of meddling in the U.S. presidential election. His comments came amid questions about the hacking of the Democratic National Committee’s computer servers, which researchers have concluded was likely the work of two Russian intelligence agencies.

Later in the news conference, when asked if he was really urging a foreign nation to hack into the private email server of Mrs. Clinton, or at least meddle in the nation’s elections, he dismissed the question. “That’s up to the president,” Mr. Trump said, before finally telling the female questioner to “be quiet — let the president talk to them.”

Mr. Trump has largely dismissed assertions that Russia was behind the Democratic committee breach as conspiracy theories — a view he reiterated again when he said the hack “is probably not Russia.”

But at the news conference at one of his Florida golf courses, as the third day of the Democratic National Convention was set to begin in Philadelphia, the Republican presidential nominee refused to unequivocally call on Vladimir V. Putin, Russia’s president, to not meddle in the United States’ presidential election.